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Thread: Inequality Application

  1. #1
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    Inequality Application

    A charter airline finds that on its Saturday flights from Philadelphia to London, all 120 seats will be sold if the ticket price is $200. However, for each $3 increase in ticket price, the number of seats sold decreases by one.

    a) Find a formula for the number of seats sold if the ticket price is P dollars.
    The answer for a is $\displaystyle -\frac{1}{3}P + \frac{560}{3}$. However, I do not see how they arrived at that answer. Any help will be greatly appreciated.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by mathgeek777 View Post
    The answer for a is $\displaystyle -\frac{1}{3}P + \frac{560}{3}$. However, I do not see how they arrived at that answer. Any help will be greatly appreciated.
    do you realize that we have a linear relationship between the number of seats filled and the ticket price? since each changes at a constant rate with respect to each other.

    lets think of this graphically so that you get the idea. we want a function of price, so put that on the x-axis. the number of seats on the y-axis. now, plot a few points:

    $\displaystyle \begin{array}{c|c} S & P \\ \hline 120 & 200 \\ 119 & 203 \\ 118 & 206 \\ . & . \\ . & . \\ . & . \end{array}$

    S is the number of seats, P is the price. S goes down by 1 when P goes up by 3

    all we need to do is find the straight line that passes through these points. can you do that?
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    Ok. I see where the $\displaystyle -\frac{1}{3}$ came from, but i still don't know how you get $\displaystyle \frac{560}{3)$
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by mathgeek777 View Post
    Ok. I see where the $\displaystyle -\frac{1}{3}$ came from, but i still don't know how you get $\displaystyle \frac{560}{3}$
    the equation of a line in the slope-intercept form is $\displaystyle y = mx + b$, where $\displaystyle m$ is the slope and $\displaystyle b$, is the y-intercept.

    given any two points on the line, $\displaystyle (x_1, y_1)$ and $\displaystyle (x_2,y_2)$, we can find the equation of the line as follows:

    $\displaystyle m = \frac {y_2 - y_1}{x_2 - x_1}$

    By the point-slope form, plug in the values of $\displaystyle m$, $\displaystyle x_1$ and $\displaystyle y_1$ into:

    $\displaystyle y - y_1 = m(x - x_1)$

    and solve for $\displaystyle y$ to get it into the slope-intercept form.

    alternatively, we could plug the points into $\displaystyle y = mx + b$ and solve for $\displaystyle b$, since it would be the only unknown at this point

    here of course, your y is S and your x is P
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  5. #5
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    Hello, mathgeek777!

    A charter airline finds that on its Saturday flights from Philadelphia to London,
    all 120 seats will be sold if the ticket price is $200.
    However, for each $3 increase in ticket price, the number of seats sold decreases by one.

    a) Find a formula for the number of seats sold if the ticket price is $\displaystyle P$ dollars.

    The answer is: $\displaystyle -\frac{1}{3}P + \frac{560}{3}$

    Let $\displaystyle x$ = number of $3-increases in the price of a ticket.

    Then the price is: .$\displaystyle P \:=\:200 + 3
    x\quad\Rightarrow\quad x \:=\:\frac{P - 200}{3}\;\;{\color{blue}[1]}$

    . . and the number of seats sold is: .$\displaystyle S \;=\;120 - x\;\;{\color{blue}[2]}$


    Substitute [1] into [2]: . $\displaystyle S \;=\;120 - \frac{P-200}{3} \;=\;\frac{360}{3} - \frac{P-200}{3} \;=\;\frac{-P + 560}{3}$

    Therefore: . $\displaystyle S \;=\;-\frac{1}{3}P + \frac{560}{3}$

    Thanks from birthdayalex
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    Re: Inequality Application

    Quote Originally Posted by Jhevon View Post
    the equation of a line in the slope-intercept form is $\displaystyle y = mx + b$, where $\displaystyle m$ is the slope and $\displaystyle b$, is the y-intercept.

    given any two points on the line, $\displaystyle (x_1, y_1)$ and $\displaystyle (x_2,y_2)$, we can find the equation of the line as follows:

    $\displaystyle m = \frac {y_2 - y_1}{x_2 - x_1}$

    .....
    That thread is very old, but I dont understand something and I have a question why the slope is $\displaystyle m = \frac {-1}{3}$ ?

    I used the formula of the slope and I used also the values of the table then:
    $\displaystyle m = \frac {203-200}{119-120}$ the answer is $\displaystyle m=-3$ and not $\displaystyle \frac {-1}{3}$


    Sincerely thanks,
    Last edited by birthdayalex; Jun 7th 2018 at 10:42 AM.
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  7. #7
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    Re: Inequality Application

    Quote Originally Posted by birthdayalex View Post
    That thread is very old, but I dont understand something and I have a question why the slope is $\displaystyle m = \frac {-1}{3}$ ?

    I used the formula of the slope and I used also the values of the table then:
    $\displaystyle m = \frac {203-200}{119-120}$ the answer is $\displaystyle m=-3$ and not $\displaystyle \frac {-1}{3}$
    For some reason, JHevon's chart is mapping seats to prices when the problem is mapping prices to seats. He switched the x- and y- axes (giving you slopes that are inverses of the ones in the problem). The slope should be:

    $$m = \dfrac{119-120}{203-200} = -\dfrac{1}{3}$$
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